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sin (x_1 + x_2 + ...+ x_n) < sin x_1 + sin x_2 + ,,,+ sin x_n

Source: Polish MO second round 1954 p6

August 29, 2024
inequalitiestrigonometry

Problem Statement

Prove that if x1,x2,…,xn x_1, x_2, \ldots, x_n are angles between 0∘ 0^\circ and 180∘ 180^\circ , and n n is any natural number greater than 1 1 , then sin⁡(x1+x2+…+xn)<sin⁡x1+sin⁡x2+…+sin⁡xn. \sin (x_1 + x_2 + \ldots + x_n) < \sin x_1 + \sin x_2 + \ldots + \sin x_n.
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